276 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
The greatest exponent for mod P/' is 
— 1 if = 1, 
Oif > 2. 
The greatest exponent for mod Qp' is 
— 1 if f^i = 
(a/ + — 1) if [Xi > 1, 
and the greatest exponent for mod m is the L.C.M, of these separate greatest 
exponents. 
Primitive roots exist when the greatest exponent is equal to <I> {m), and 
$ (m) = fh (1 + iy . ^ (Pi^‘) . ^ (Py^) . . . <t) cp (Q/^) . . . 
and 
cp (1 + iy = 1 if /c = 1 
= 2''-' if K > 1, 
cp (P^^i) = p^2 _ 1 if = 1 
= Pi3(^i-i)(Pj3- 1) if Xi > 2, 
= a/ + - 1 if Pi = 1 
= W + ^ — !)• 
The only cases in which the greatest exponent can be equal to «P {ra) are those in 
which the separate exponents are each equal to the (P of the modulus they refer to 
and are also co-prime. 
Hence we have the foliov/ing moduli possessed of primitive roots :— 
(1) The moduli 1 + 7, (1 -f if, (1 + ^f \ 
(2) Any pure prime and (1 +7) X any pure prime ; 
(3) A power of a mixed prime and (1 X a power of a mixed prime. 
Example .—To find the exponent of 3 + 2?’ for mod 1000. 
1000 = 23. 53 = - (1 + if (2 4- if (1 + 2/)3. 
The exp of 3 + 27 for mod (1 -f 2y’ is 4. 
3 -f 27 = 1 (mod 2 -j- 7), 
and therefore has exponent 1. 
(3 -f- 27)^ ^ 1 [mod (2 -f 7)"] 
tlierefore 
